Gårding's inequality

Gårding's inequality

In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator. The inequality is named after Lars Gårding.

tatement of the inequality

Let Ω be a bounded, open domain in "n"-dimensional Euclidean space and let "H""k"(Ω) denote the Sobolev space of "k"-times weakly-differentiable functions "u" : Ω → R with weak derivatives in "L"2. Assume that Ω satisfies the "k"-extension property, i.e., that there exists a bounded linear operator "E" : "H""k"(Ω) → "H""k"(R"n") such that ("Eu")|Ω = "u" for all "u" in "H""k"(Ω).

Let "L" be a linear partial differential operator of even order "k", written in divergence form

:(L u)(x) = sum_{0 leq | alpha |, | eta | leq k} (-1)^ mathrm{D}^{alpha} left( A_{alpha eta} (x) mathrm{D}^{eta} u(x) ight),

and suppose that "L" is uniformly elliptic, i.e., there exists a constant "θ" > 0 such that

:sum_ int_{Omega} A_{alpha eta} (x) mathrm{D}^{alpha} u(x) mathrm{D}^{eta} v(x) , mathrm{d} x

is the bilinear form associated to the operator "L".

Application: the Laplace operator and the Poisson problem

As a simple example, consider the Laplace operator Δ. More specifically, suppose that one wishes to solve, for "f" ∈ "L"2(Ω) the Poisson equation

:egin{cases} - Delta u(x) = f(x), & x in Omega; \ u(x) = 0, & x in partial Omega; end{cases}

where Ω is a bounded Lipschitz domain in R"n". The corresponding weak form of the problem is to find "u" in the Sobolev space "H"01(Ω) such that

:B [u, v] = langle f, v angle mbox{ for all } v in H_{0}^{1} (Omega),

where

:B [u, v] = int_{Omega} abla u(x) cdot abla v(x) , mathrm{d} x,:langle f, v angle = int_{Omega} f(x) v(x) , mathrm{d} x.

The Lax-Milgram lemma ensures that if the bilinear form "B" is both continuous and elliptic with respect to the norm on "H"01(Ω), then, for each "f" ∈ "L"2(Ω), a unique solution "u" must exist in "H"01(Ω). The hypotheses of Gårding's inequality are easy to verify for the Laplace operator Δ, so there exist constants "C" and "G" ≥ 0

:B [u, u] geq C | u |_{H^{1} (Omega)}^{2} - G | u |_{L^{2} (Omega)}^{2} mbox{ for all } u in H_{0}^{1} (Omega).

Applying the Poincaré inequality allows the two terms on the right-hand side to be combined, yielding a new constant "K" > 0 with

:B [u, u] geq K | u |_{H^{1} (Omega)}^{2} mbox{ for all } u in H_{0}^{1} (Omega),

which is precisely the statement that "B" is elliptic. The continuity of "B" is even easier to see: simply apply the Cauchy-Schwarz inequality and the fact that the Sobolev norm is controlled by the "L"2 norm of the gradient.

References

* cite book
author = Renardy, Michael and Rogers, Robert C.
title = An introduction to partial differential equations
series = Texts in Applied Mathematics 13
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 2004
pages = 356
isbn = 0-387-00444-0
(Theorem 8.17)


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